Much as Math 71 is concerned with the study of groups and rings,
Math 81 involves the study of fields and more precisely extensions of
fields. The theory which is developed along the way provides solutions
to a number of classical problems, as well as introduces tools which
intertwine the study of fields with that of groups, rings, and vector
spaces.

In a sense much of algebra owes its heritage to questions of finding
solutions to equations. We all learn in grade school that equations
like x^6+ x^5 + ... + x + 1 = 0, have all of their roots in the
complex numbers, but what is the "smallest" field containing all its
roots, and does that field have arithmetic interest? Is there a
formula like the quadratic formula which can describe the roots of
all polynomials in terms of radicals and algebraic operations?

Short of resorting to the use of the real or complex numbers which
are more germane to analysis, we shall learn how to construct the
smallest field containing the rational numbers in which any given
polynomial (or all possible polynomials) have their roots. We will
show the impossibility of several classical compass and straightedge
constructions: trisection of an angle, creating a square with the same
area as a circle, or even constructing a cube whose volume is twice
that of a given cube. We shall also show that there is no formula for
finding the roots of equations of degree at least five using radicals
and algebraic operations.

All of these are very pleasing results,
owing largely to the distinction between being unable to show
something is possible and being able to prove that it is not.